Theorem un0.1 42399

Description: ⊤ is the constant true, a tautology (see df-tru 1542). Kleene's "empty conjunction" is logically equivalent to ⊤. In a virtual deduction we shall interpret ⊤ to be the empty wff or the empty collection of virtual hypotheses. ⊤ in a virtual deduction translated into conventional notation we shall interpret to be Kleene's empty conjunction. If 𝜃 is true given the empty collection of virtual hypotheses and another collection of virtual hypotheses, then it is true given only the other collection of virtual hypotheses. (Contributed by Alan Sare, 23-Apr-2015.) (Proof modification is discouraged.) (New usage is discouraged.)

Hypotheses

Ref	Expression
un0.1.1	⊢ (   ⊤   ▶   𝜑   )
un0.1.2	⊢ (   𝜓   ▶   𝜒   )
un0.1.3	⊢ (   (   ⊤   ,   𝜓   )   ▶   𝜃   )

Assertion

Ref	Expression
un0.1	⊢ (   𝜓   ▶   𝜃   )

Proof of Theorem un0.1

Step	Hyp	Ref	Expression
1		un0.1.1	. . . 4 ⊢ (   ⊤   ▶   𝜑   )
2	1	in1 42191	. . 3 ⊢ (⊤ → 𝜑)
3		un0.1.2	. . . 4 ⊢ (   𝜓   ▶   𝜒   )
4	3	in1 42191	. . 3 ⊢ (𝜓 → 𝜒)
5		un0.1.3	. . . 4 ⊢ (   (   ⊤   ,   𝜓   )   ▶   𝜃   )
6	5	dfvd2ani 42203	. . 3 ⊢ ((⊤ ∧ 𝜓) → 𝜃)
7	2, 4, 6	uun0.1 42398	. 2 ⊢ (𝜓 → 𝜃)
8	7	dfvd1ir 42193	1 ⊢ (   𝜓   ▶   𝜃   )

Syntax hints:  ⊤wtru 1540  (   wvd1 42189  (   wvhc2 42200

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This theorem depends on definitions:  df-bi 206  df-an 397  df-tru 1542  df-vd1 42190  df-vhc2 42201

This theorem is referenced by:  sspwimpVD  42539